Question

Little Italy Pizza charges \(\$ 12.00\) for a 12 -inch diameter sausage pizza and \(\$ 14.00\) for a 14 -inch diameter sausage pizza. Write the pseudocode algorithm that calculates and displays how much each of these earns the establishment per square inch of pizza sold. (Hint: you will need to first calculate how many square inches there are in each pizza.)

Short answer

Answer: Little Italy Pizza earns approximately: - \(\frac{1}{3\pi}\) dollars per square inch for the 12-inch sausage pizza - \(\frac{2}{7\pi}\) dollars per square inch for the 14-inch sausage pizza

Step-by-step solution

Identify the given information

We are given the prices and diameters of the following sausage pizzas: - 12-inch diameter pizza costs $12.00 - 14-inch diameter pizza costs $14.00

Calculate the area of each pizza

To find the area of each pizza, we need to use the formula for the area of a circle: Area = π × (radius)^2 First, we need to convert the diameters to radii. The radius is half of the diameter, so: - Radius of 12-inch pizza: 12/2 = 6 inches - Radius of 14-inch pizza: 14/2 = 7 inches Now, we can calculate the areas: - Area of 12-inch pizza: π × (6)^2 = 36π square inches - Area of 14-inch pizza: π × (7)^2 = 49π square inches

Calculate the earnings per square inch for each pizza

To find the earnings per square inch, we will divide the price of each pizza by its respective area: - Earnings per square inch for 12-inch pizza: \(\frac{12}{36\pi} = \frac{1}{3\pi}\) - Earnings per square inch for 14-inch pizza: \(\frac{14}{49\pi} = \frac{2}{7\pi}\)

Display the results

The final step is to display the earnings per square inch for each pizza: - Earnings per square inch for 12-inch pizza: \(\frac{1}{3\pi}\) dollars - Earnings per square inch for 14-inch pizza: \(\frac{2}{7\pi}\) dollars

Key concepts

Geometry

Geometry plays a crucial role in understanding the design and measurement of shapes and spaces. When it comes to calculating areas, geometry provides the essential formulas needed to determine the sizes of different shapes, like circles. In our exercise, the focus is on a circular pizza, and its area is a critical part of the earnings calculation.
To calculate the area of a circle, we use the formula: \[ \text{Area} = \pi \times (\text{radius})^2 \] The radius is half of the diameter, which is the length from one side of the circle to the other through the center.
Given a pizza's diameter, we convert it to a radius by dividing by two. Therefore, understanding these basic geometric conversions is a vital step towards solving problems like this.

Area Calculation

Calculating area is a key concept in many mathematical problems. In this context, we want to determine how much pizza surface we are dealing with. The area of a pizza tells us how many square inches it covers, which is vital when determining cost efficiency.
For the 12-inch pizza:

• Convert the diameter to radius: 12 inches / 2 = 6 inches
• Find the area: \( \pi \times 6^2 = 36\pi \) square inches

For the 14-inch pizza:

• Radius is 7 inches, calculated as 14 inches / 2
• Area is \( \pi \times 7^2 = 49\pi \) square inches

Understanding these calculations helps to figure out the next steps in the problem, particularly how much money each square inch is generating.

Earnings Calculation

When it comes to determining the earnings per square inch, you need to factor in both the price of the pizza and its area. This gives the restaurant insights into how profitably each pizza size sells relative to its surface area.
For the 12-inch pizza, the calculation is handled as follows:

• Price: \(12.00
• Area: \(36\pi \) square inches
• Calculate earnings per square inch: \( \frac{12}{36\pi} = \frac{1}{3\pi}\)

Similarly, for the 14-inch pizza:

• Price: \)14.00
• Area: \(49\pi \) square inches
• Earnings per square inch: \(\frac{14}{49\pi} = \frac{2}{7\pi}\)

These calculations show which pizza might be more cost-effective or offer better overall earnings per square inch.

Step-by-step Solution

Breaking down each mathematical problem into step-by-step solutions ensures clarity and better understanding. This process allows us to tackle complicated problems systematically, making them more manageable.
Step 1: Identify all necessary information. Know the diameters and costs of each pizza before you start with any calculations.
Step 2: Calculate the area of each pizza to understand the total surface area. Use the formula \( \pi \times (\text{radius})^2 \).
Step 3: Compute the earnings per square inch by dividing the cost of each pizza by its area.
Step 4: Display the results, making sure that the results are clear and error-free.Using these steps, you will be able to solve similar problems efficiently, by focusing on the process rather than jumping straight to solutions.